The magnitude and direction angle of v are ||v|| = 12 and θ = 60°. Express v in terms of i and j.

Key Concepts

Magnitude of a Vector

The magnitude of a vector, denoted as ||v||, represents its length in a given space. In this case, ||v|| = 12 indicates that the vector has a length of 12 units. Understanding magnitude is crucial for visualizing the size of the vector and is often the first step in vector analysis.

Direction Angle

The direction angle θ of a vector indicates its orientation relative to a reference axis, typically the positive x-axis. Here, θ = 60° means the vector is positioned at a 60-degree angle from the x-axis. This angle is essential for decomposing the vector into its components along the x and y axes.

Vector Components

Vector components are the projections of a vector along the coordinate axes, usually represented as i (x-axis) and j (y-axis). To express vector v in terms of i and j, we use the formulas v_x = ||v|| * cos(θ) and v_y = ||v|| * sin(θ). For θ = 60°, this results in specific values for the x and y components, allowing for a complete representation of the vector.